Sections of Lie group actions and a theorem by M. Newman

Details Sections of Lie group actions and a theorem by M. Newman Sections of Lie group actions and a theorem by M. Newman

2006-08-15

arxiv.org/abs/math/0608387v1

Proceedings of the conference ``Fundamental mathematics today'' devoted to the 10-th annivesary of the Independent University

Mathematics

Dynamical Systems

10 pages, this is a translation from russian of the paper published in Proceedings of the conference ``Fundamental mathematics

Scientific paper

Let M be a smooth finite-dimensional manifold, G a Lie group, and \Phi:G x M --> M a smooth action. Consider the following mapping $\phi: C^{\infty}(M,G) --> C^{\infty}(M,M)$, defined by $\phi(\alpha)(x) = \alpha(x)\cdot x$, for $\alpha\in C^{\infty}(M,G)$ and $x\in M$. In this paper we describe the structure of inverse images of elements of $C^{\infty}(M,M)$ under $\phi$ for $\dim G=1$, i.e. when $G$ is either $R^1$ or $S^1$. As an application we obtain a new proof of the well-known theorem by M. Newman concerning the interior of the fixed point set of a Lie group action.

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